I added the "arithmetic-progression" tag because the solution for n=1 is any arithmetic progression, giving a difference set with cardinality $2m-1$. So in some sense the higher-dimensional solutions generalize arithmetic progressions.
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Keenan PepperSep 20 '11 at 5:02

Please also mention on math.SE that you cross posted.
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quidSep 20 '11 at 13:14

1 Answer
1

A basic inequality proved in 1987 by Freiman, Heppes, and Uhrin ("A lower estimation for the cardinality of finite difference sets in $R^n$", Number theory, Vol. I (Budapest, 1987), 125–139, Colloq. Math. Soc. János Bolyai, 51, North-Holland, Amsterdam, 1990) is that $|S-S|\ge(n+1)|S|-n(n+1)/2$. A number of improvements have been obtained since then; in particular, Stanchescu ("On finite difference sets", Acta Math. Hungarica 79 (1998), no. 1-2, 123–138) showed that for $n=3$ one has $|S-S|\ge 4.5|A|-9$, with an explicit description of those sets $S$ for which equality is attained.

You can recover much more starting with these two papers and their MathReviews.